A model of diffusion and mass flow of water in cylindrical membrane systems with application to plant roots

Summary Measurements of the radial diffusion of tritiated water, combined with axial and radial flow in an artificial cylindrical membrane, are examined with the aid of a mathematical model. The results are used to assess how far measurement of diffusion of labeled water in plant roots may throw light on pathways of movement of water and on barriers to flow.     

Determination of the membrane permeability coefficient and the reflection coefficient by the two-dimensional laminar flow model for intestinal perfusion experiments

We performed single perfusion experiments in the small intestine of rats in order to prove that the two-dimensional laminar flow model is suitable to determine the membrane permeability coefficient and the reflection coefficient. We used progesterone as an aqueous-diffusion-limited drug, urea as a membrane transport-limited drug and the tritiated water as an intermediate substance. The membrane permeability coefficient for progesterone was calculated to be 3.6 X 10(-4) cm/s. This value did not change when the thickness of the aqueous diffusion layer was altered by increasing the perfusion rate 10-fold. It was directly demonstrated that the two-dimensional laminar flow model was suitable to analyze the data of intestinal perfusion experiments. Membrane permeability coefficients for urea and tritiated water were determined to be 3.4 X 10(-5) cm/s and 8.9 X 10(-5) cm/s, respectively. In the presence of water absorption with the hypotonic perfusion solution, the reflection coefficient for urea was 0.84. This value is thought to be theoretically reasonable, suggesting the usefullness of the two-dimensional laminar flow model to obtain the reflection coefficient in the intestinal membrane.     

A Continuum Mechanical Approach to the Flow Equations for Membrane Transport: I. Water Flow

A concept is presented for modeling flows through membranes using continuum mechanics. Viscous interactions (due to velocity gradients) are explicitly incorporated and position-dependent local water-membrane interactions are taken into account before obtaining slab averages. This is in distinction to other treatments where strictly one-dimensional force balance equations are written using slab average friction coefficients which are really composite functions of local interactions. It is shown that the viscous and other frictional interactions do not simply form linear combinations in the solutions to the equations of motion. Flow profiles for pressure-driven flows ranging from Poiseuille's flow to “diffusion” flow are obtained depending on the strength and extent of the water-membrane interaction. The model is also applied to self-diffusion flows and the measurement of “equivalent pore size.” It is shown that for a fixed pore size the ratio of filtration flow to self-diffusion flow for equal driving forces is able to vary over a wide range depending on the water-membrane interaction.     

Water transport across isolated term human amnion

Summary The hydrodynamic permeability of normal term human amnion is measured using pressure-driven bulk flows. The permeability coefficient is found to vary widely, variations between tissues taken from different subjects being significantly greater than those from samples taken from one subject. No correlation is observed between this coefiicient and either tissue thickness or the diffusional permeability coefficient measured using tritiated water; it is, however, found to be very sensitive to epithelial damage.The results indicate that the bulk transport of water through amnion is largely controlled by the amniotic epithelium alone. This contrasts with water diffusion which is a function of total membrane thickness. The two permeability coefficients cannot therefore be employed to formulate an equivalent pore model of the whole tissue. An equivalent pore model of the epithelial layer only is considered and the results assessed in the light of other evidence bearing on the structure of amnion. It is concluded that the epithelial layer is intersected by a large number of pores with radius 10 to 30 Å, and a smaller number of much broader pores.     

The influence of flow rates and flow distributions on diffusion controlled transport across the isolated guinea pig placenta

Summary The influence of flow rates and flow distributions on diffusion controlled transport across the isolated guinea pig placenta was investigated.It was shown that the transport of antipyrine and tritiated water could be explained by simple diffusion and concluded that the maternal and fetal circulations are directed countercurrently. Antipyrine and tritiated water were also used to calculate which part of the maternal and fetal circulations were directed away from the exchange area. One of the models developed by Meschia (1) was changed to account for existing shunts in the maternal and fetal circulations.     

An analysis of long-distance water transport in the soybean stem using H215O

The lateral water movement in the intact stem of a transpiring soybean plant was analyzed quantitatively by a real-time measurement system utilizing labeled water, H(2)(15)O and gamma ray detectors. A large volume of water escaping from xylem vessels during its transport was detected. The escape of water was not influenced by evaporation from the stem surface or mass flow in the sieve tubes. It was assumed that the total amount of water transported through xylem vessels was kept almost completely constant along the internode. As a result, most of the escaped water was found to re-enter the xylem vessels, i.e. water exchange occurred. The analysis of radiographs of tritiated water suggested that the self-diffusion effect of water was strong for lateral water movement, although another driving force besides thermal motion was included in the process, and that the process was also affected by the water permeability of the plasma membrane. An analysis based on a mathematical model showed that the net volume of water which escaped from xylem vessels was not dependent on the transpiration rate of the plant.     

An NMR study of the temperature dependence of the coefficient of water self-diffusion through lipid bilayer membranes

The temperature dependence of the coefficient of water self-diffusion through plane-parallel lipid multilayers of the phospholipid dioleoylphosphatidylcholine oriented on a glass support has been studied in the temperature range of 20-60 degrees C by the method of NMR with magnetic field pulse gradient. The values of the coefficients of transbilayer water diffusion are by four orders of magnitude less than for bulky water and ten times less than the coefficients of lateral diffusion of the lipid under the same conditions. The temperature dependence of the coefficient of water diffusion is described by the Arrhenius law with an apparent activation energy of about 41 kJ/mol, which far exceeds the activation energy for the diffusion of bulky water (18 kJ/mol). The experimental data were analyzed using a "dissolving-diffusion" model, by simulating the passage of water through membrane channels, and by analyzing the exchange of water molecules in states with different modes of translation mobility, including pore channels and bilayer "defects". Each of the approaches used made it possible to take the significance of bilayer permeability for the apparent energy of activation of water diffusion into account and estimate the energies of activation of water diffusion in the hydrophobic moiety of the bilayer, which were found to be close to the values for bulky water. The coefficients of water diffusion in the system under examination and the coefficients of permeation of water through the bilayer were estimated, and the effect of bilayer "defects" on the coefficients of water diffusion along and across bilayers was studied.     

On the large error introduced in the estimate of the density of membrane pores from permeability measurements when diffusion in "unstirred layer" within the cells is disregarded

After the development of the "black lipid membrane" techniques, studies of the permeability of labeled water and nonelectrolytes across these artificial membranes have yielded permeability constants comparable in magnitude to those obtained from tracer studies of living cell membranes. This general agreement has affirmed the belief that the living cell membranes are indeed closely similar to these bilayer phospholipid membranes. In this report, we draw attention to a hidden assumption behind such comparisons made: the assumption that labeled material passing through the cell membrane barriers instantly reaches diffusion equilibrium inside the cell. The permeability constants to labeled water (and nonelectrolytes) across lipid layers were obtained using setups in which the lipid membrane was sandwiched between aqueous compartments both of which were vigorously stirred. In studies of permeability of living cell membranes only the outside solution was stirred, the intracellular water remained stationary. Yet the calculations of permeability constants of the cell membrane were made with the tacit assumption, that once the labeled materials pass through the cell membrane, they were instantly mixed with the entire cell contents as if a stirrer operating at infinite speed had been present inside the cells. Ignoring this unstirred condition of the intracellular water, in fact, lumped all the real-life delay due to diffusion in the cytoplasm and added it to the resistance to diffusion of the membrane barrier. The result is an estimated membrane permeability to labeled water (and nonelectrolytes) many times slower than it actually is. The present report begins with a detailed analysis of a specific case: tritiated water diffusion from giant barnacle muscle fibers and two non-living models, one real, one imagined.(ABSTRACT TRUNCATED AT 250 WORDS)     

Self-diffusion of polymers in cartilage as studied by pulsed field gradient NMR

Pulsed field gradient (PFG) nuclear magnetic resonance (NMR) was used to investigate the self-diffusion behaviour of polymers in cartilage. Polyethylene glycol and dextran with different molecular weights and in different concentrations were used as model compounds to mimic the diffusion behaviour of metabolites of cartilage. The polymer self-diffusion depends extremely on the observation time: The short-time self-diffusion coefficients (diffusion time Delta approximately 15 ms) are subjected to a rather non-specific obstruction effect that depends mainly on the molecular weights of the applied polymers as well as on the water content of the cartilage. The observed self-diffusion coefficients decrease with increasing molecular weights of the polymers and with a decreasing water content of the cartilage. In contrast, the long-time self-diffusion coefficients of the polymers in cartilage (diffusion time Delta approximately 600 ms) reflect the structural properties of the tissue. Measurements at different water contents, different molecular weights of the polymers and varying observation times suggest that primarily the collagenous network of cartilage but also the entanglements of the polymer chains themselves are responsible for the observed restricted diffusion. Additionally, anomalous restricted diffusion was shown to occur already in concentrated polymer solutions.     

Transient effects on the initial rate of oxygenation of red blood cells

The rate-controlling process in the oxygenation of red blood cells is investigated using a Roughton-like model for oxygen diffusion and reaction with hemoglobin. The mathematical equations describing the model are solved using two independent techniques, numerical inversions of the Laplace transform of the equations and numerical solutions via an implicit-explicit finite difference form of the equations. The model is used to re-examine previous theoretical models that incorporate either a red cell membrane that is resistive to oxygen diffusion or an unstirred layer of water surrounding the cell. Although both models have been postulated to be equivalent, the results of the computer simulations demonstrate significant differences between the two models in the rate of oxygenation of the red cells, depending upon the values chosen for the diffusion coefficient for O₂ in the membrane and the thickness of the water layer. The difference is apparently due to differences in the induction and transient periods of the water layer model relative to the membrane model.     

The State of Water in Human and Dog Red Cell Membranes

The apparent activation energy for the water diffusion permeability coefficient, P_d, across the red cell membrane has been found to be 4.9 ± 0.3 kcal/mole in the dog and 6.0 ± 0.2 kcal/mole in the human being over the temperature range, 7° to 37°C. The apparent activation energy for the hydraulic conductivity, L_p, in dog red cells has been found to be 3.7 ± 0.4 kcal/mole and in human red cells, 3.3 ± 0.4 kcal/mole over the same temperature range. The product of L_p and the bulk viscosity of water, η, was independent of temperature for both dog and man which indicates that the geometry of the red cell membrane is not temperature-sensitive over our experimental temperature range in either species. In the case of the dog, the apparent activation energy for diffusion is the same as that for self-diffusion of water, 4.6–4.8 kcal/mole, which indicates that the process of water diffusion across the dog red cell membrane is the same as that in free solution. The slightly, but significantly, higher activation energy for water diffusion in human red cells is consonant with water-membrane interaction in the narrower equivalent pores characteristic of these cells. The observation that the apparent activation energy for hydraulic conductivity is less than that for water diffusion across the red cell membrane is characteristic of viscous flow and suggests that the flow of water across the membranes of these red cells under an osmotic pressure gradient is a viscous process.     

Computing the self-diffusion coefficient for TIP4P water

A molecular dynamics study was made for the TIP4P model of liquid water. Thermal dependences of water density and radial distribution functions were calculated for model verification. Different methods were used to calculate the self-diffusion coefficient, and assessed for sensitivity to molecular system size and trajectory length. The Green-Kubo formula deriving the diffusion coefficient from the velocity autocorrelation function is preferable in short MD simulations with a high sampling rate, whereas the Einstein equation for diffusion is the method of choice in long simulations. The latter approach yields more stable and reliable results, especially at very short times and for a small number of molecules, if the diffusion coefficient is estimated not from the limit ratio of mean squared displacement to time, but from the slope of the time plot of mean squared displacement.     

PFG-NMR measurements of the self-diffusion coefficients of water in equilibrium poly(HEMA-co-THFMA) hydrogels

The self-diffusion coefficients for water in a series of copolymers of 2-hydroxyethyl methacrylate, HEMA, and tetrahydrofurfuryl methacrylate, THFMA, swollen with water to their equilibrium states have been studied at 310 K using PFG-NMR. The self-diffusion coefficients calculated from the Stejskal-Tanner equation, D(obs), for all of the hydrated polymers were found to be dependent on the NMR storage time, as a result of spin exchange between the proton reservoirs of the water and the polymers, reaching an equilibrium plateau value at long storage times. The true values of the diffusion coefficients were calculated from the values of D(obs) in the plateau regions by applying a correction for the fraction of water protons present, obtained from the equilibrium water contents of the gels. The true self-diffusion coefficient for water in polyHEMA obtained at 310 K by this method was 5.5 x 10(-10) m(2)s-1. For the copolymers containing 20% HEMA or more a single value of the self-diffusion coefficient was found, which was somewhat larger than the corresponding values obtained for the macroscopic diffusion coefficient from sorption measurements. For polyTHFMA and copolymers containing less than 20% HEMA, the PFG-NMR stimulated echo attenuation decay curves and the log-attenuation plots were characteristic of the presence of two diffusing water species. The self-diffusion coefficients of water in the equilibrium-hydrated copolymers were found to be dependent on the copolymer composition, decreasing with increasing THFMA content.     

Relationship of net chloride flow across the human erythrocyte membrane to the anion exchange mechanism

The parallel effects of the anion transport inhibitor DIDS (4,4'- diisothiocyanostilbene-2,2'-disulfonate) on net chloride flow and on chloride exchange suggest that a major portion of net chloride flow takes place through the anion exchange system. The "slippage" model postulates that the rate of net anion flow is determined by the movement of the unloaded anion transport site across the membrane. Both the halide selectivity of net anion flow and the dependence of net chloride flux on chloride concentration over the range of 75 to 300 mM are inconsistent with the slippage model. Models in which the divalent form of the anion exchange carrier or water pores mediate net anion flow are also inconsistent with the data. The observations that net chloride flux increases with chloride concentration and that the DIDS- sensitive component tends to saturate suggest a model in which net anion flow involves "transit" of anions through the diffusion barriers in series with the transport site, without any change in transport site conformation such as normally occurs during the anion exchange process. This model is successful in predicting that the anion exchange inhibitor NAP-taurine, which binds to the modifier site and inhibits the conformational change, has less effect on net chloride flow than on chloride exchange.     

Modeling tree water flow as an unsaturated flow through a porous medium

The electric circuit analogy has had a profound influence on how tree physiologists measure, model and think about tree water flow. For example, previous models that attempt to account for changes in saturation use the electric circuit analogy to define capacitance as the change in saturation per change in pressure. Given that capacitance is constant, this relationship implies that subjecting a block of wood to a pressure of -2.5 MPa for 2 min results in the same change in saturation as subjecting the same block to the same pressure for 2 days. Given the definition of capacitance, it is unclear how the electric circuit analogy could be used to predict changes in saturation separately from changes in pressure. The inadequacies in the electric circuit analogy discussed in this paper necessitate a new theory of tree water flow that recognizes the sapwood as being a porous medium and explicitly deals with the full implications of the unsaturated flow occurring in the sapwood. The theory proposed in this paper combines the Cohesion theory with a mathematical theory of multiphase flow through porous media. Based on this theory, both saturated and unsaturated tree water flow models are presented. Previous partial differential equation models of tree water flow based on the electric circuit analogy are shown to be mathematically equivalent to the model of saturated porous flow. The unsaturated model of tree water flow explicitly models the pressure profile and the rates of change in saturation and specific interfacial area (a measure of how the water in the unsaturated sapwood is partitioned between mobile and immobile components). The unsaturated model highlights the differences between saturated and unsaturated flow and the need to measure the variables governing tree water flow at higher spatial and temporal resolutions.     

On simultaneous transport of water and solute through plant cell membranes: Evidence for the absence of solvent drag effect and insensitivity of the reflection coefficient

The simultaneous efflux of tritiated water and ¹⁴C labelled ethanol from inner epidermal cells of the bulb scale of Allium cepa was measured with a specially designed efflux chamber. It was found that water and ethanol moved essentially independently. Rates of efflux of tritiated water and ¹⁴C ethanol were essentially the same in the presence or absence of a simultaneous influx of water. Using the same technique the efflux of tritiated water from the epidermal cells was measured during a simultaneous flow of nonlabelled ethanol. When tritiated water and ethanol moved in opposite directions, the water permeability values became slightly reduced depending upon the concentration of ethanol. When ethanol and tritiated water moved in the same direction, however, no effect on water permeability values could be detected. These results are best explained by the molecular theory of diffusion across lipid bilayer membranes, and are consistent with the above findings of lack of interaction between water and ethanol as they are transported across the cell membrane. In another study, the solute permeability coefficients (K_s) for non-electrolytes such as urea and methyl urea were measured by plasmolyzing the epidermal cells and transferring them to equimolal solutions of urea and methyl urea. This method was also used to measure the reflection coefficient (σ) for these nonelectrolytes. The K_s values for methyl urea were 16 times greater than the ones for urea. The values of σ for both of these solutes, however, were very close to 1. Using the K_s data available in the literature for the subepidermal cells of the Pisum sativum stem basis, the σ values were calculated for malonamide, glycerol, methyl urea, ethyl urea, dimethyl urea, and formamide. Again the K_s values for these nonelectrolytes varied by several orders of magnitude, whereas all σ values were found to be close to 1. These findings point out that σ is an insensitive parameter and that K_s, the solute permeability constant, has to be used for characterizing solute transport through the membrane. The present study shows that fast (e.g. ethanol, formamide) as well as slowly permeating molecules do not interact with water as they are transported across the cell membrane. Aqueous pores for the simultaneous transport of water and solutes, therefore, are absent in the plant cell membranes investigated here.     

Steady-state methods for the study of placental exchange.

Published and new data on steady-state exchange of tracers and oxygen are characterized by marked species differences. When the placenta is treated as an ideal diffusion cell of unknown vessel geometry and permeability, the exchange characteristics of oxygen can be used to prove that the exchange of tracers such as acetylene, nitrous oxide, tritiated water and antipyrine is entirely flow limited. The recorded patterns of transfer of flow-limited tracers reveal that some placentas are as effective as counter-current exchangers whereas others mimic the behavior of the less effective types of exchangers. These species differences in apparent geometry are unrelated to the histologic nature of the barrier. The patterns of flow-limited transfer are so similar to those of oxygen transfer that the diffusion resistance to oxygen must be small. The exchange of diffusion-limited (hydrophilic) tracers mimics exchange across capillary membranes in some species and exchange across cell membranes in others. These species differences in diffusion-limited transfer are clearly related to the histologic nature of the barrier and are independent of vessel geometry.     

Is the Cell Membrane a Universal Rate-Limiting Barrier to the Movement of Water between the Living Cell and Its Surrounding Medium?

With the use of a newly introduced technique, the "influx profile analysis," we studied the diffusion of tritiated water in and out of frog ovarian eggs at 25°C. The results show that the rate-limiting step in the exchange of labeled water is not permeation through the cell membrane but diffusion in the bulk of the intracellular water.